In principle, the atom-dimer scattering length can take any value. However, for a deep atom-atom potential, such as for the ground-state strontium dimer [1]}, there is a universality of the three-body physics that sets \(\kappa _*=0.226(2)/l_{\mathrm {vdW}}\)  [2]}. Here, \(l_{\mathrm {vdW}}=\left({2\mu C_6}/{\hbar ^2}\right)^{1/4}/2=74.6\)  \(a_0\)  is the van der Waals length associated with the \(C_6\)  coefficient of the long-range Sr\(_2\)  ground-state potential. We use \(C_6=3.03(1) \times 10^{-76}\)   J m\(^6\)  found from a fit of potential parameters to spectroscopic data [1]}, which is consistent with a recent ab initio calculation [4]}. This yields \(\kappa _*=5.72\times 10^7\)  m\(^{-1}=(330\,a_0)^{-1}\) . Equation (REF ) then predicts \(a_{\text{ad}}=6.4\, a_{86}\) , which leads to a small density-dependent frequency shift parameter of \(\chi _n=50\,\mathrm {Hz}/(10^{12}\,\mathrm {cm}^{-3})\) . A numerical calculation including a finite-range correction for the atom-atom interaction [5]} results in \(a_{\text{ad}}=3.5\, a_{86}\)  and \(\chi _n=-90\,\mathrm {Hz}/(10^{12}\,\mathrm {cm}^{-3})\) . Thus, a very small shift is expected for the densities used here. We incorporate \(\chi _n=0\pm 90 \,\mathrm {Hz}/(10^{12}\,\mathrm {cm}^{-3})\)  as a set parameter in our model of the spectrum, where we set the systematic uncertainty to reflect the spread of theory predictions. This uncertainty will be significant for our determination of the unperturbed halo binding energy.
